Jet variability in simple models

Jet variability in simple models Zachary Erickson October 2, 2014 Abstract Zonal jets are a feature common to rotating planets, and are prevelant on earth in the atmosphere and the Southern Ocean (SO), as well as on other planets in our solar system. These jets are typically formed through a convergence of momentum in regions with eddy activity. Jet formation and persistence under steady-state conditions have been well studied, but the effects of time-varying parameters are not well known. Here we use a simple two-layer quasi-geostrophic (QG) model to consider how changing the friction strength over time affects the distribution of jets. We demonstrate a hysteresis between the number of jets in the domain and the friction in our model. A commonly used scaling for jet spacing, the Rhines length, is not accurate for our simulations. Finally, we consider the effect of eddies on the movement of jets in our domain through Reynolds stresses. 1 Introduction The rotation of planets constrains fluid motion such that meridional deviations are discouraged and zonal flow is dominant. Zonal jets are common features in planetary atmospheres. They can be clearly seen in the banded structure of Jupiter, and are also present in the earth s atmosphere. Zonal motion is ubiquitous throughout all ocean basins (Maximenko, et al., 2005), although other effects, such as the presence of continental boundaries or other external forcings, may mask this phenomenon (Rhines, 1994). In the Southern Ocean (SO), fluid motion is unconstrained by continents and can therefore form circumpolar flows. The Antarctic Circumpolar Current (ACC) is the dominant feature between 35 60, but can also be sub-divided into various fronts which are typically associated with individual jets (Orsi, et al., 1995; Belkin and Gordin, 1996). Zonal jets can be maintained by a large-scale meridional temperature gradient, such as the subtropical jets associated with the Hadley cell, or by eddies, as is the case in the mid-latitude atmospheric jets. The latter case is the result of a convergence of momentum into regions of eddy development, leading to an eastward flow in the stirring region and westward flow elsewhere (Vallis, 2007). The stability of jets is well documented (e.g. Panetta, 1993), but various observations and models have called into question the permanence of individual jets or jet structures. In the atmosphere, an analysis of global climate models shows a narrowing and poleward motion of the mid-latitude jet in the southern hemisphere, which is thought to strengthen 1

Ekman transport and eddy activity in the ACC (Fyfe and Saenko, 2006). Harnik, et al. (2014) examine an anomalous merging of two jet streams in 2010 due to an unusually strong meridional temperature gradient; such merging events could have large implications for weather patterns. In the ocean, the shedding of rings from the Agulhas retroflection, which propagate westward and mix Indian ocean water into the Atlantic, is the result of an instablity in the jet structure in the SO east of Africa (Lutjeharms and van Ballegooyen, 1988). In addition, jets in the ACC exhibit instability, as they merge and dissipate spatially and temporally throughout the region (Sokolov and Rintoul, 2007). The formation of zonal jets is a result of conservation of potential vorticity (PV). PV can be expressed as P V = 2 ψ + f, (1) h where 2 is the Laplacian ( 2 x 2 + 2 y 2 ), ψ is the streamfunction, such that (u, v) = ( ψ y, ψ x ), f is the planetary vorticity, and h is the height of the fluid layer. Conservation of PV means a change in f can be compensated by a change in the relative vorticity ( 2 ψ), creating a restoring force which causes zonal flow to dominate. We choose the simplest model, to be described in Section 2, that captures the dynamics in the system. Broadly speaking, we solve the layer-wise equations D Dt ( 2 ) ψ i + βy = δ i2 fric. hypν, (2) h i where i is the layer, D/Dt consists of the partial time derivative and advective terms ( / t + u ), β is the latitudinal gradient of f, δ i2 fric. is friction present only in the bottom layer, and hypν is the hyperviscosity. We use this model to examine how the jet structure reacts to changes in the bottom friction. In this formulation, friction acts as a baroclinic element in the system. Although this model is intended mostly for oceanic applications, we do not include a wind stress term; however, this would also act as a baroclinic feature, meaning the core dynamics of the system can be expressed through bottom friction alone. The rest of this paper is organized as follows. The model is introduced in Section 2, and a discussion of jet formation follows in Section 3. The variable we choose to vary is the bottom drag (κ). Steady-state equilibrium conditions for three values of κ are identified and characterized at the end of Section 3. In the next section, we vary κ between these values and demonstrate a state of hysteresis between κ and the number of jets in our domain. We then briefly discuss the role of Reynolds stresses and PV fluxes in the movement of jets in the domain before concluding in Section 5. 2 Model We use a two-layer, quasi-geostrophic (QG) model with a rigid lid on a beta plane (Philips, 1954). We assume a large-scale pressure gradient (i.e. caused by a decrease 2

in averaged solar heating from the equator to the poles) which induces a large-scale baroclinic structure in the mean zonal velocity (U), where we define U = U 1 = U 2 (subscripts refer to the upper and lower layers, respectively). PV conservation is expressed in our model as, q 1,t + J(ψ 1, q 1 ) + Uq 1,x + (β + Uλ 2 )ψ 1,x = ν 8 q 1 q 2,t + J(ψ 2, q 2 ) Uq 2,x + (β Uλ 2 )ψ 2,x = ν 8 q 2 κ 2 ψ 2, (3a) (3b) where q 1 = 2 ψ 1 + 1 2 λ 2 (ψ 2 ψ 1 ) q 2 = 2 ψ 2 + 1 2 λ 2 (ψ 1 ψ 2 ), (4a) (4b) J(a, b) = (a x b y a y b x ) is the horizontal Jacobian, λ is the Rossby radius of deformation, ν is the hyperviscosity parameter, κ is the bottom friction, and q i and ψ i are small-scale perturbations. The Rossby radius λ is calculated by g H/2f0 2, where g is the reduced gravity, H is the average layer depth (2H is the total depth), and f 0 is the average planetary vorticity. In our model the mean shear between the layers, 2U, is a fixed quantity, meaning the large-scale PV gradient is constant. Eddies typically act to reduce this gradient. Because in our model this is not possible, the fixed shear theoretically acts as an infinite supply of energy that could be used for jet or eddy production. In our simulations, the damping effect of bottom friction (along with hyperviscosity) allows our model to equilibrate. We avoid the influence of boundaries in x and y by making our model doubly periodic. Longitudinal periodicity is justified because the jets we are interested in modeling are circumpolar. Latitudinal periodicity stems from the idea that there exists a large-scale global gradient in pressure, and we are interested in modeling a small portion in the middle of this gradient, far from any boundary effects. While this greatly simplifies the system, lack of boundaries does not mean that the domain size is unimportant. Specifically, L/L jet, where L is the lenth of the (square) domain and L jet is the average jet spacing, is constrained to the set of integers because there cannot exist a fractional jet. In our simulations, L/L jet O(10). This rounding effect means that a change in a system from n to n ± 1 jets may be trivial. 3 Jet formation and characteristics 3.1 Jet formation We start each simulation with negligible q 1 and q 2. The nonlinear terms in (3) are therefore small and can be ignored. If we neglect both dissipative effects, our system of 3

Figure 1: a: Contours of growth rate s during the linear stage as a function of zonal and meridional wavenumbers k and l, calculated at κλ/u = 0 and βλ 2 /U = 0.02. b: Contours of maximum growth rate s during the linear stage as a function of β and κ, calculated at k = λ and l = 0. equations becomes q 1,t + Uq 1,x + (β + Uλ 2 )ψ 1,x = 0 q 2,t Uq 2,x + (β Uλ 2 )ψ 2,x = 0. (5a) (5b) Using the ansatz (ψ i = ˆψ i e i(kx+ly ωt) ), we find that the growth rate s = Im(ω) is at a maximum for (k, l) (0, λ), where k and l are the zonal and meridional wavenumbers, respectively (Figure 1a). Shear in U between layers acts as a source of energy to waves of all k and l, and the waves which have the maximum growth rate dominate. This stage is generally known as the elevator mode, since k = 0 means it has no structure in the meridional direction (Figure 2a). Friction acts to reduce the total amount of kinetic energy in the system. Interestingly, this does not mean an increase in κ always decreases the amount of eddy kinetic energy in the initial growth stage. If we include the frictional term in our analysis, we find that, for most values of β, the maximum growth rate, which stays relatively constant at (k, l) (0, λ) decreases as κ increases (Figure 1b). However, when β becomes large, an increased bottom drag actually increases the growth rate, presumably because friction constrained to the lower layer increases the baroclinic component of the system, which can act as a source of eddy kinetic energy (Thompson and Young, 2007). 4

Figure 2: a: Contours of q 1 (upper layer) during the linear growth stage, also known as the elevator mode (tu/λ = 30). b: Beginning of the transition state, where non-linear terms become important (tu/λ = 40). Note the change in color scale. c: Sample q 1 field in steady state (tu/λ = 200). q has a zonal structure, but many eddies are also present. The magnitude of q 1 is somewhat less than during the transition state. All snapshots are taken from a simulation with βλ 2 /U = 0.5, κλ/u = 0.1, and ν/(uλ 7 ) = 0.25. An increase in β monotonically decreases the maximum growth rate. Low β means there is only a small meridional change in the planetary vorticity. Therefore, fluid parcels can be displaced in the y direction with little restoring force. High β, conversely, designates a large change in the planetary rotation with a change in latitude, which acts to more heavily constrain flow to the x direction. As the elevator mode grows, q increases and the non-linear terms become important. There is a short transition state, in which the eddy kinetic energy reaches its maximum, and zonal motions become dominant. After time, the model reaches a steady-state equilibrium at (k, l) = (k jet, 0), which describes zonal jets (Figure 2b,c). Solving a priori for the jet spacing, L jet, is non-trivial. One common scaling is known as the Rhines length (Rhines, 1975). For a single layer over flat bathymetry, (1) becomes P V = 2 ψ + βy (6) where the second and third terms are the advective and planetary terms and dissipative effects are neglected. From this we can determine the Rhines length scale U eddy L Rh β, (7) where U eddy is typically taken as the root-mean-squared eddy velocity. Assuming the proportionality constant is O(1), L2 β U 1 implies that the beta-term dominates and the non-linear advective terms are negligible, leading to wave-like, sinusoidal behavior. Conversely, when L2 β U 1, the non-linear advective term dominates. The beta term 5

restricts meridional motion, and the advective term, through the strength of the eddies, mixes fluid meridionally. At some point (the proportionality constant) these effects cancel each other out, giving the length scale associated with jets. 3.2 PV staircase The total PV, Q, in the top layer is given by Q 1 = q 1 + (β + Uλ 2 )y. (8) Initially, q 1 is small and Q 1 is essentially a linear function of y. The elevator stage contains only small perturbations in q 1, but as jets form q 1 increases. Eastward jets are bounded by cyclonic eddies (q > 0) on the north flank and anti-cyclonic eddies (q < 0) to the south. Westward jets are opposite, leading to westward jets forming in areas of PV homogenization and eastward jets being associated with large latitudinal gradients of PV (Dritschel and McIntyre, 2008). 3.3 Equilibrium states The important three parameters we control are the planetary gradient in vorticity (β), the bottom drag (κ), and the hyperviscosity (ν). Non-dimensionalized, these are expressed as β = βλ2 U, κ = κλ U, and ν = ν. We expect the hyperviscous parameter Uλ 7 to have a negligible influence on the dynamics of the system (Panetta, 1993; Thompson and Young, 2006). Correct selection of ν is model-based and dependent on the model resolution. This parameter is responsible for small-scale dissipation in the system. As the resolution increases ν can be decreased because finer-scale structures are possible. Our model uses 256 256 resolution, and we empirically find ν = 0.25 to work well. A small number of tests at 512 resolution and the same ν did not give qualitatively different results. Thompson and Young (2007) recently used a similar model to study the effects of β and κ on the eddy diffusivity of temperature, D τ. This diffusivity is related to the eddy strength, which we expect to affect jet spacing through (7). They found that the dependence of D τ on κ varied strongly with respect to β. For β 0.7, D τ was essentially independent of κ, and the dependence increased as β moved away from 0.7. We tested a variety of conditions before finding a suitable value of β such that an adjustment in κ would effect a large change in jet spacing. Our chosen equilibrated states, with β = 0.5 and κ = [0.1, 0.02, 0.005], are shown in Figure 3. 4 Results and discussion 4.1 Hysteresis We start with the equilibrated states in Figure 3 and vary the bottom drag within κ = [0.1, 0.02, 0.005] to evaluate how the jets react. One such simulation is shown in Figure 4, where we initialize the jet structure at κ = 0.1 and cycle stepwise through our 6

Figure 3: Snapshot of zonal jet structure in our three systems, with β = 0.5, ν = 0.25, and κ = a: 0.1, b: 0.02, and c: 0.005 after they have equilibrated (tλ/u = 2000). frictional parameter space. Similar models have shown that jets are temporally quite stable (Panetta, 1993). Our simulation shows that jets also have considerable stability to changes in friction, remaining in the same structure during a 5-fold decrease in κ from 0.1 to 0.02. However, when the drag is further decreased to κ = 0.005 the jet structure breaks and the jets reform into a new equilibrium. Similarly, as the drag is increased, the initial step only strengthens the jets, and another step is required to re-form them. Friction acts to oppose fluid motion. We might expect that a lower frictional term would increase the energy available to the eddies, causing them to become stronger and increasing the jet spacing, as shown in (7) and Figure 3. We define u i,jet = u i, 2 (9a) u eddy = (u 1 u 1,jet ) 2 + v1 2 + (u 2 u 2,jet ) 2 + v2 2, (9b) where the overbar denotes a horizontal average (in x and y). These values are shown in Figure 4c,d. The first step-change in bottom drag initially increases u eddy, but that extra energy is quickly taken up by u jet, and the jets intensify. Initially, the second drop in drag follows the same formula, as u jet increases once more. However, as if a threshold value was crossed, some of the energy available stays in u eddy, which increases, peaking as the jet structure begins to break up at at tλ/u = 4250. As the jets reform at a larger spacing, eddy energy decreases, which is counterintuitive from the Rhines scale. Weaker jet activity evidently has an effect on eddy strength. The low drag may also have decreased the baroclinicity of the system enough to decrease the eddy kinetic energy from that component. We see that as the friction increases again at tλ/u = 6000 both u jet and u eddy increase, even though the jet structure does not change. It is only when κ is increased back to 0.1 that the jets again re-form themselves. 7

Figure 4: a: Hovmoller of non-dimensionalized u x over time. Drag changes stepwise as indicated in b. u eddy and u jet (for the upper layer) are shown in c and d. A step-change in drag is not very physical. Many of the same experiments were performed with a slow, linear change in κ, which did not qualitatively change the observed effects. In one simulation, shown in Figure 5, we initialized the system at κ = 0.1 for 2000 model days, slowly decreased κ to 0.005 over 3000 model days, and held it there for an additional 2000 model days. The jets kept their position until tλ/u 5100, which was after κ had leveled out at 0.005. We find that while the peak in u eddy also occurs at tλ/u 5100, the increase in eddy velocity begins at about tλ/u = 4630, when κ 0.016. Throughout the decrease in drag, and indeed until the jet structure breaks up, u jet is increasing, meaning it is constantly taking excess energy out of u eddy. However, there appears to be a tipping point at κ 0.016 where u jet is no longer able to take up all of this energy. u eddy grows, reaching its maximum as the jets reorganize themselves, implying that the change in eddy strength caused the jets to reform as expected by (7). 4.2 Reynolds stresses and PV fluxes We now consider how jets reorganize. A jet can be considered as an accumulation of zonal momentum in the mean flow. The zonal energy equation in the upper layer is D Dt KE 1 = u 1 (u 1 v 1 ) y, (10) where the overbar denotes a zonal and time average. The term on the right hand side involves the divergence of the Reynolds stress, u v. Reynolds stresses arise from the 8

Figure 5: a: Hovmoller diagram of non-dimensionalized u x over time for a simulation which linearly decreases κ from 0.1 to 0.005, as shown in b. Also shown is u eddy and u jet over time (c,d). Note that the large spike in u eddy begins as the drag is still decreasing but the peak, which corresponds to the break-up in the jet structure, occurs after the drag has stabilized. interplay between eddies and jets. An isotropic eddy has u v = 0. When eddies are sheared by an eastward mean flow, eddies on the southern flank of the jet develop positive Reynolds stresses, whereas u v on the northern flank is negative. Thus, (u v ) y < 0 and energy flows from the eddies into the mean flow. The divergence of the Reynolds stress can also be related to the meridional flux of PV. In the upper layer, friction is non-existent and, if we neglect hyperviscosity, q 1,t + J(ψ 1, q 1 ) = 0. (11) Taking the zonal and time mean, we find that, in a domain with x periodicity, (v 1 q 1 ) y = q 1,t. (12) Thus, in steady state the eddy meridional flux of PV is constant. From (4a), it can be shown that (u 1 v 1 ) y = v 1 q 1 1 2 λ 2 ψ 1,x ψ 2. (13) This is a variation on the Taylor-Bretherton identity. In the lower layer, we must account for friction. Neglecting the hyperviscous term, q 2,t + J(ψ 2, q 2 ) = κ ψ 2. (14) 9

We take the zonal and time mean, and recover (v 2 q 2 ) = q 2,t + κu 2,y. (15) The eddy meridional flux of PV in the lower layer is therefore directly related to the mean flow. Jets are largely barotropic, meaning u 1 u 2 (this is less true in the cores of the jets). The meridional flux of PV in the lower layer could thus act as a transfer of energy between the layers. We can more clearly show this transfer by using (4b) to find (u 2 v 2 ) y = v 2 q 2 + 1 2 λ 2 ψ 1,x ψ 2. (16) The term ψ 1,x ψ 2 acts to transfer energy between the layers. Combining (13) and (16), v 1 q 1 + (u 1 v 1 ) y + v 2 q 2 + (u 2 v 2 ) y = 0, (17) where in steady state v 1 q 1 is a constant, v 2 q 2 = f(u 2), and (u 1 v 1 ) y and (u 2 v 2 ) y act directy on u 1 and u 2, respectively (although the Reynolds stresses in the lower layer are typically quite small). Figure 6 shows the relationship between non-dimensionalized u 1 and u 1 (u 1 v 1 ) y for a stationary jet and a jet which is moving northward. For the stationary jet, the Reynolds stress divergence term is at a maximum at the center of the jet. In this case, the eddies serve to strengthen and narrow the jet. The moving jet, however, has a Reynolds stress divergence which is slightly offset in the direction of motion of the jet, showing that Reynolds stresses play a role in moving the jet. In addition, the Reynolds stress divergence term is negative on the southern flank of the migrating jet, meaning here energy is being taken out of the mean flow and put into the eddies. 5 Conclusion and future work We used a simple quasi-geostrophic model to demonstrate an inherent hysteresis between jet spacing and bottom friction. The interaction between eddies and the mean flow is highly significant. Decreasing bottom drag increases the strength of eddies, which can eventually become strong enough to force jets to migrate and merge. Eddies can themselves move jets through the meridional gradient of Reynolds stresses, according to (10). We find a connection between the meridional flux of PV and the divergence of the Reynolds stresses in both layers in (17). Using simplifications discussed in the previous section, in steady-state we can re-write (17) as (u 1 v 1 ) y κu 2 + C, (18) where C is a constant composed of v 1 q 1 and the constant of integration involved in solving (15) and we neglect any contribution by (u 2 v 2 ) y as small. Note that when q i,t 0, additional complexity enters through (12) and (15). We have found occasions in our simulations where threshold values of κ seem to have been reached which cause the jets to reform. We see these in the evolution of u eddy, 10

Figure 6: Blue lines show the non-dimensionalized zonal jet u/u; green lines show the contribution to the mean flow from the non-dimensionalized eddy component, u(u v ) y L jet /U 3 /10. Dashed lines are values averaged over the zonal domain size and 500 time units during which the κ is constant and the jets are in steady state. Solid lines are values averaged over the zonal domain size and 20 time units during with κ is linearly increasing from 0.02 to 0.1 and the chosen jet is migrating northward (upward). The y axis is set so that y = 0 is the center of each jet, meaning y = 0 changes position in the domain as the jet moves. which does not much vary with changing drag outside of these threshold conditions. Future work involves a careful study of these values of κ which seem to greatly affect the system. Little time has been spent analyzing the second half of Figure 4, where the drag increases. A curious effect between eddy strength and jets was observed, where a small number of jets in the domain coincides with low u eddy, in defiance of the Rhines scaling (7). One way forward may be through (18). As κ decreases, the divergence of the Reynolds stress in the upper layer decreases as well, meaning there is less transfer between the jets and the eddies according to (10). In addition, a lower u 2 also decreases this transfer term. Jets are largely barotropic, so small u 2 implies small u 1 and a weak jet, which is what we see during the κ = 0.005 stage of Figure 4. A better understanding of how eddies and jets feed on and off each other is therefore a future goal. Finally, it is unclear to what extent conclusions derived from this highly simplified model relate to the real world. We plan to use satellite data from AVISO (Archiving, Validation, and Interpretation of Satellite Oceanographic Data) to detect zonal jets and upper layer Reynolds stress divergences in the ocean, especially where jets are strong and variable, such as leeward of the Kerguelen plateau in the SO and in the Kuroshio extension in the northwest Pacific. AVISO uses satellite altimetry to generate temporal maps of sea surface height (SSH), from which geostrophic flow can be determined. From this flow the jet structure and Reynolds stresses in the upper layer can be approximated, and we can investigate to what extent (10) holds. 11

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